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Clifford Algebra
|arXiv://On the “equivalence” of the Maxwell and Dirac equations> André Gsponer "It is shown that Maxwell’s equation cannot be put into a spinor form that is equivalent to Dirac’s equation." "This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell’s and Dirac’s equations into each other." |UGlasgow://Investigation of electromagnetism in a real Dirac algebra> - Stephen Leary, 2007 "The history of algebras presented here begins with an introduction to the history of numbers themselves and a description of a famous problem in mathematics known as the Pythagorean catastrophe." "Negative numbers and zero were concepts yet to be imagined for the Pythagoreans. They had hoped that all geometric systems could be described using rational numbers i.e. numbers of the form a/b where a &b are positive integers. A simple example, however, can be used to show that this hope was in vain. Take a simple right angle triangle in Euclidean geometry where the sides are of length unity." "The word “Algebra” comes from a book written in Arabic which revolutionised the way mathematics was conducted. The book, entitled “Al-jebr w’al-mugabalah”, was written by Abu Ja’far Ben Musa also known as al-Khowarizmi around 825 AD11. This book is widely regarded as being the origin of algebra and the basis for the, eventual, wider acceptance of zero as a number through-out Europe. The first use of the word “algebra” in English was by the Welsh mathematician and textbook writer, Robert Recorde12." "In the 19th century British mathematicians took the lead in the study of algebra. Attention turned to many “algebras”; that is various sorts of mathematical objects ( i.e. vectors, matrices, transformations, etc.) and various operations which could be carried out upon these objects. Thus the scope of “algebra” was expanded to the study of algebraic form and structure and was no longer limited to ordinary systems of numbers" "Perhaps the most significant breakthrough in mathematics is the development of non-commutative algebras. These are algebras in which significance is attached to the ordering of the operation of multiplication. The first example of such an algebra is the quaternions which Hamilton first wrote down on the 16th of October 1843 by scoring them onto the side of Brougham bridge in Dublin, less than a year before Grassmann’s exterior algebra was published. As a progression from this, in 1878 William Kingdon Clifford published his paper entitled “Applications of Grassmann’s extensive algebra”13 in which a non-commutative geometric product is presented. Although Clifford’s motivations appear to be purely academic his work was influenced by Riemann and Lobachevsky. This paper is the seminal paper for the class of algebras now regarded by modern mathematics as Clifford algebras." "In the early 20th century Paul Dirac, in an effort to find a first order form of the relativistic Schr ̈odinger equation, constructed a non-commutative algebra which fullfiled specific criteria14. In finding his non-commuting ‘spin’ quantities, Dirac rediscovered an algebra with many of the same properties as the four dimensional Clifford algebras. Dirac appears to have been completely unaware of Clifford’s work. As a consequence, algebras with properties common to both are sometimes referred to as Dirac-Clifford algebras8." |David Hestenes://REAL DIRAC THEORY> "The Dirac theory is completely reformulated in terms of Spacetime Algebra, a real Clifford Algebra characterizing the geometrical properties of spacetime. This eliminates redundancy in the conventional matrix formulation and reveals a hidden geometric structure in the theory. Among other things, it reveals that complex numbers in the Dirac equation have a kinematical interpretation, with the unit imaginary identified as the generator of rotations in a spacelike plane representing the direction of electron spin. Thus, spin and complex numbers are shown to be inextricably related in the Dirac Theory. This leads to a new version of the zitterbewegung, wherein local circular motion of the electron is directly associated with the phase factor of the wave function. In consequence, the electron spin and magnetic moment can be attributed to the zitterbewegung, and many other features of quantum mechanics can be explained as zitterbewegung resonances." Category:Mathematics Category:Quantum Mechanics Category:Wave Theory Category:Spin